Momentum-Accelerated Richardson(m) and Their Multilevel Neural Solvers

TL;DR

Solving large-scale linear systems from PDE discretizations remains challenging due to parameter variability and scale. The authors propose Richardson($m$) neural solvers that learn long-step weights $\bm \omega$ via a meta-network $\text{Meta}(\theta;\bm \mu)$, and enhance them with momentum (including Nesterov) and preconditioning, achieving faster convergence than classic Richardson variants. They further integrate these solvers into multilevel frameworks, yielding Richardson($m$)-FNS for anisotropic diffusion and NAG-Richardson($m$)-WANS for Helmholtz problems, with alternating-optimization training across multiple meta-networks. The results show competitive or superior performance to Chebyshev methods without eigenvalue computations, along with strong robustness across PDE parameters and grid sizes, highlighting the practical potential of neural-augmented multilevel solvers for parametric PDEs.

Abstract

Recently, designing neural solvers for large-scale linear systems of equations has emerged as a promising approach in scientific and engineering computing. This paper first introduce the Richardson(m) neural solver by employing a meta network to predict the weights of the long-step Richardson iterative method. Next, by incorporating momentum and preconditioning techniques, we further enhance convergence. Numerical experiments on anisotropic second-order elliptic equations demonstrate that these new solvers achieve faster convergence and lower computational complexity compared to both the Chebyshev iterative method with optimal weights and the Chebyshev semi-iteration method. To address the strong dependence of the aforementioned single-level neural solvers on PDE parameters and grid size, we integrate them with two multilevel neural solvers developed in recent years. Using alternating optimization techniques, we construct Richardson(m)-FNS for anisotropic equations and NAG-Richardson(m)-WANS for the Helmholtz equation. Numerical experiments show that these two multilevel neural solvers effectively overcome the drawback of single-level methods, providing better robustness and computational efficiency.

Figures (4)

• Figure 1: A schematic of a Meta$(\bm \theta;\bm \mu)$ network, i.e., an FNN.
• Figure 2: Computational flow of Richardson(m)-FNS.
• Figure 3: Network architecture of Meta-$T$ and Meta-$\tilde{\Lambda}$.
• Figure 4: Computational flow of NAG-Richardson(m)-WANS.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3